Fourier Analysis and Partial Differential Equations
Authors
Description
It was with the creation of differential and integral calculus in the 17th century that the study of differential equations began. Research in this area was guided by its applications to particle mechanics. These applications included the use of Newton’s three laws of dynamics and the law of universal gravitation. From then on it was possible to obtain differential equations that represent the phenomena under study.
Since partial differential equations are mathematically difficult to solve, the author shows and develops some methods. These are indispensable for showing how to solve some partial differential equations present in mathematical physics problems. These include the Fourier method in the first chapter. A theory of Fourier series, problems of heat conduction in a bar, problems for the one-dimensional wave equation, the Fourier transform and the Dirichlet problem are also discussed.
Target audience
Higher education
Name: Fourier Analysis and Partial Differential Equations
Author(s): e Djairo Guedes de Figueiredo
Pages: 292
Publication: IMPA, 2018
ISBN: 978-85-244-0428-3
Edition: 5
CONTENTS
1 Why study Fourier series?
1 Heat conduction in a bar
2 Mathematical formulation of the heat conduction problem
Exercises
2 Fourier series
1 Periodic functions
2 Uniform convergence
3 Fourier coefficients
4 Fourier series
5 Fourier series of even and odd functions
6 Calculation of some Fourier series
7 Integration of Fourier series
8 Estimation of Fourier coefficients
9 Complex form of Fourier series
10 Parseval’s identity
11 Historical note
Exercises
2 Convergence of Fourier series
1 Classes of functions considered
2 Point convergence of the Fourier series
3 Riemann-Lebesque lemma
4 Point convergence of the Fourier series (continuation)
5 Bessel Inequality
6 Cauchy-Schwarz and Minkowski inequalities
7 Uniform convergence of the Fourier series
8 Dirac nuclei
9 Weierstrass approximation theorem
10 Fejér’s theorem
11 Parseval’s identity
12 Functions of limited variation
13 Gibbs phenomenon
14 Isoperimetric problem
15 Historical note
Exercises
4 Heat equation
1 Heat conduction: bar with ends held at 0ºC
2 Heat conduction: bar subject to other lateral conditions
3 Inhomogeneous boundary conditions
4 Inhomogeneous heat equation
5 Heat conduction in an inhomogeneous bar
6 Uniqueness of PVIF solution (4.1)
7 Soil temperature variations
Exercises
5 Wave equation
1 Vibrating string equation
2 Fourier series solution
3 Vibrating string energy
4 Harmonics, frequency, amplitude
5 Fingered string
6 Forced vibrations. Resonance
7 Infinite string
8 Semi-infinite string
9 Transmission lines
10 Longitudinal vibrations of an elastic bar
11 Generalized Sobolev solutions**
Exercises
6 Fourier transforms and applications
1 For motivation
2 Definition of the Fourier transform
3 L-space and the Fourier transform in L
4 Convolution product
5 Plancherel’s theorem
6 Poisson’s sum formula and the heat equation
7 Cauchy’s problem for the heat equation
8 Heat conduction in a semi-infinite bar infinite bar
Appendix – Functions represented by integrals
Exercises
Answers and suggestions to the exercises
Bibliography
Index